There are several types of conventional techniques for migrating seismic data. The objective of all migration techniques is to generate a display of processed seismic data in which the position of each seismic reflection indicated by the display accurately represents the position within the earth of whatever caused the reflection.
In one conventional class of seismic migration techniques, known as "KF-migration" (or "FK-migration") techniques, the seismic data to be processed are first Fourier-transformed, then the Fourier-coefficients are migrated, and thereafter, the migrated coefficients are inverse-Fourier-transformed. Examples of such KF-migration methods are described in the following papers: R. H. Stolt, "Migration by Fourier Transform," Geophysics, V. 43, No. 1 (February, 1978), pp. 23-48; J. Gazdag, "Wave Equation Migration with the Phase-Shift Method," Geophysics, V. 43, No. 7 (December, 1978), pp. 1342-1351; and J. Gazdag, "Wave Equation Migration with the Accurate Space Derivative Method," Geophysical Prospecting, V. 28, (1980), pp. 60-70.
A drawback of conventional KF-migration techniques of the type described in the Stolt paper is that they require that the relationship between depth of a reflector in the earth and the seismic wave traveltime to and from the reflector must be represented as a single fixed ratio. In other words, those methods require that the velocity of seismic waves in the earth be represented as a constant.
The Gazdag (1978) technique has the disadvantage that it is time-consuming and expensive to implement, for a number of reasons, including the reason that the number of operations required to migrate each seismic trace in accord with the Gazdag (1978) technique is proportional to N.sup.2, where N is the number of data points (or samples) comprising the trace. The Gazdag (1980) technique, too, has the disadvantage of being time-consuming and expensive to implement.
The present invention represents an improvement over conventional KF-migration techniques because it facilitates efficient migration of seismic data without requiring that the velocity of the associated seismic waves be represented as a constant. Rather, the invention requires only that the earth be represented as a sequence of uniform horizontal layers, each layer having a characteristic velocity, or ratio of depth increment to traveltime. The number of operations required to migrate a single trace in accord with the present invention is proportional to N log N, where N is the number of data points (or samples) comprising the trace.